subproduct systems - uni-bielefeld.de · 2010. 1. 15. · orr moshe shalit and baruch solel 1...

Documenta Math. 801

Subproduct Systems

Orr Moshe Shalit and Baruch Solel 1

Received: July 1, 2009

Revised: August 12, 2009

Communicated by Joachim Cuntz

Abstract. The notion of a subproduct system, a generalization ofthat of a product system, is introduced. We show that there is anessentially 1 to 1 correspondence between cp-semigroups and pairs(X,T ) where X is a subproduct system and T is an injective subprod-uct system representation. A similar statement holds for subproductsystems and units of subproduct systems. This correspondence is usedas a framework for developing a dilation theory for cp-semigroups. Re-sults we obtain:(i) a ∗-automorphic dilation to semigroups of ∗-endomorphisms overquite general semigroups;(ii) necessary and sufficient conditions for a semigroup of CP maps tohave a ∗-endomorphic dilation;(iii) an analogue of Parrot’s example of three contractions with noisometric dilation, that is, an example of three commuting, contrac-tive normal CP maps on B(H) that admit no ∗-endomorphic dilation(thereby solving an open problem raised by Bhat in 1998).Special attention is given to subproduct systems over the semigroupN, which are used as a framework for studying tuples of operatorssatisfying homogeneous polynomial relations, and the operator alge-bras they generate. As applications we obtain a noncommutative(projective) Nullstellensatz, a model for tuples of operators subject tohomogeneous polynomial relations, a complete description of all rep-resentations of Matsumoto’s subshift C∗-algebra when the subshiftis of finite type, and a classification of certain operator algebras –including an interesting non-selfadjoint generalization of the noncom-mutative tori.

2000 Mathematics Subject Classification: 46L55, 46L57, 46L08,47L30.

1B.S. was supported in part by the US-Israel Binational Foundation and by the Fund forthe Promotion of Research at the Technion.

Documenta Mathematica 14 (2009) 801–868

802 Orr Shalit and Baruch Solel

Keywords and Phrases: Product system, subproduct system,semigroups of completely positive maps, dilation, e0-dilation, ∗-automorphic dilation, row contraction, homogeneous polynomialidentities, universal operator algebra, q-commuting, subshift C∗-algebra.

Contents

Introduction 802

Part 1. Subproduct systems and cp-semigroups 8101. Subproduct systems of Hilbert W ∗-correspondences 8102. Subproduct system representations and cp-semigroups 8123. Subproduct system units and cp-semigroups 8194. ∗-automorphic dilation of an e0-semigroup 8215. Dilations and pieces of subproduct system representations 826

Part 2. Subproduct systems over N 8346. Subproduct systems of Hilbert spaces over N 8347. Zeros of homogeneous polynomials in noncommutative variables 8408. Universality of the shift: universal algebras and models 8449. The operator algebra associated to a subproduct system 84810. Classification of the universal algebras of q-commuting tuples 85311. Standard maximal subproduct systems with dimX(1) = 2 and

dimX(2) = 3 85712. The representation theory of Matsumoto’s subshift C∗-algebras 861References 866

Introduction

Motivation: dilation theory of CP0-semigroups. We begin by describ-ing the problems that motivated this work.Let H be a separable Hilbert space, and let M ⊆ B(H) be a von Neumannalgebra. A CP map on M is a contractive, normal and completely positivemap. A CP0-semigroup on M is a family Θ = {Θt}t≥0 of unital CP maps onM satisfying the semigroup property

Θs+t(a) = Θs(Θt(a)) , s, t ≥ 0, a ∈M,

Θ0(a) = a , a ∈ B(H),and the continuity condition

limt→t0〈Θt(a)h, g〉 = 〈Θt0(a)h, g〉 , a ∈ M, h, g ∈ H.


Subproduct Systems 803

A CP0-semigroup is called an E0-semigroup if each of its elements is a ∗-endomorphism.Let Θ be a CP0-semigroup acting onM, and let α be an E0-semigroup actingon R, where R is a von Neumann subalgebra of B(K) and K ⊇ H . Denotethe orthogonal projection of K onto H by p. We say that α is an E0-dilationof Θ if for all t ≥ 0 and b ∈ R(0.1) Θt(pbp) = pαt(b)p.

In the mid 1990’s Bhat proved the following result, known today as “Bhat’sTheorem” (see [9] for the caseM = B(H), and also [40, 15, 29, 6] for differentproofs and for the general case):

Theorem 0.1. (Bhat). Every CP0-semigroup has a unique minimal E0-dilation.

A natural question is then this: given two commuting CP0-semigroups, canone simultaneously dilate them to a pair of commuting E0-semigroups? In [43]the following partial positive answer was obtained2:

Theorem 0.2. [43, Theorem 6.6] Let {φt}t≥0 and {θt}t≥0 be two strongly com-muting CP0-semigroups on a von Neumann algebra M⊆ B(H), where H is aseparable Hilbert space. Then there is a separable Hilbert space K containing Hand an orthogonal projection p : K → H, a von Neumann algebra R ⊆ B(K)such that M = pRp, and two commuting E0-semigroups α and β on R suchthat

φs ◦ θt(pbp) = pαs ◦ βt(b)pfor all s, t ≥ 0 and all b ∈ R.In other words: every two-parameter CP0-semigroup that satisfies an addi-tional condition of strong commutativity has a two-parameter E0-dilation. Thecondition of strong commutativity was introduced in [48]. A precise definitionwill not be given here. The main tools in the proof of Theorem 0.2,and also in some of the proofs of Theorem 0.1, were product sys-tems of W∗-correspondences and their representations. In fact, theonly place in the proof of Theorem 0.2 where the assumption of strong com-mutativity is used, is in the construction of a certain product system. Moreabout that later.In [10], Bhat showed that given a pair of commuting CP maps Θ and Φ onB(H), there is a Hilbert space K ⊇ H and a pair of commuting normal ∗-endomorphisms α and β acting on B(K) such that

Θm ◦ Φn(pbp) = pαm ◦ βn(b)p , b ∈ B(K)for all m,n ∈ N (here p denotes the projection of K onto H). Later on Solel,using a different method (using in fact product systems and their representa-tions), proved this result for commuting CP maps on arbitrary von Neumannalgebras [48]. Neither one of the above results requires strong commutativity.

2The same result was obtained in [42] for nonunital semigroups acting on M = B(H).



In light of the above discussion, and inspired by classical dilation theory [47, 49],it is natural to conjecture that every two commuting (not necessarily stronglycommuting) CP0-semigroups have an E0-dilation, and in fact that the same istrue for any k commuting CP0-semigroups, for any positive integer k. However,the framework given by product systems seems to be too weak to prove this.Trying to bypass this stoppage, we arrived at the notion of a subproduct system.

Background: from product systems to subproduct systems. Prod-uct systems of Hilbert spaces over R+ were introduced by Arveson some 20years ago in his study of E0-semigroups [3]. In a few imprecise words, a prod-uct system of Hilbert spaces over R+ is a bundle {X(t)}t∈R+ of Hilbert spacessuch that

X(s+ t) = X(s)⊗X(t) , s, t ∈ R+.We emphasize immediately that Arveson’s definition of product systems re-quired also that the bundle carry a certain Borel measurable structure, but wedo not deal with these matters here. To every E0-semigroup Arveson associateda product system, and it turns out that the product system associated to anE0-semigroup is a complete cocycle conjugacy invariant of the E0-semigroup.Later, product systems of Hilbert C∗-correspondences over R+ appeared (seethe survey [46] by Skeide). In [15], Bhat and Skeide associate with everysemigroup of completely positive maps on a C∗-algebra A a product system ofHilbert A-correspondences. This product system was then used in showing thatevery semigroup of completely positive maps can be “dilated” to a semigroup of∗-endomorphisms. Muhly and Solel introduced a different construction [29]: toevery CP0-semigroup on a von Neumann algebraM they associated a productsystem of Hilbert W∗-correspondences overM′, the commutant of M. Again,this product system is then used in constructing an E0-dilation for the originalCP0-semigroup.Product systems of C∗-correspondences over semigroups other than R+ werefirst studied by Fowler [21], and they have been studied since then by manyauthors. In [48], product systems over N2 (and their representations) were stud-ied, and the results were used to prove that every pair of commuting CP mapshas a ∗-endomorphic dilation. Product systems over R2+ were also central to theproof of Theorem 0.2, where every pair of strongly commuting CP0-semigroupsis associated with a product system over R2+. However, the construction of theproduct system is one of the hardest parts in that proof. Furthermore, thatconstruction fails when one drops the assumption of strong commutativity, andit also fails when one tries to repeat it for k strongly commuting semigroups.On the other hand there is another object that may be naturally associatedwith a semigroup of CP maps over any semigroup: this object is the subproductsystem, which, when the CP maps act on B(H), is the bundle of Arveson’s“metric operator spaces” (introduced in [4]). Roughly, a subproduct system ofcorrespondences over a semigroup S is a bundle {X(s)}s∈S of correspondencessuch that

X(s+ t) ⊆ X(s)⊗X(t) , s, t ∈ S.



See Definition 1.1 below. Of course, a difficult problem cannot be made easyjust by introducing a new notion, and the problem of dilating k-parameterCP0-semigroups remains unsolved. However, subproduct systems did alreadyprovide us with an efficient general framework for tackling various problemsin operator algebras, and in particular it has led us to a progress toward thesolution of the discrete analogue of the above unsolved problem.While preparing this paper we learned that Bhat and Mukherjee have alsoconsidered subproduct systems over the semigroup R+ ([14]). They called itinclusion systems and used it to compute the index of certain product systems.This paper consists of two parts. In the first part we introduce subproductsystems over general semigroups, show the connection between subproduct sys-tems and cp-semigroups, and use this connection to obtain three main resultsin dilation theory of cp-semigroups. The first result is that every e0-semigroupover a (certain kind of) semigroup S can be dilated to a semigroup of ∗-automorphisms on some type I factor. The second is some necessary conditionsand sufficient conditions for a cp-semigroup to have a (minimal) ∗-endomorphicdilation. The third is an analogue of Parrot’s example of three contractionswith no isometric dilation, that is, an example of three commuting, contractivenormal CP maps onB(H) that admit no ∗-endomorphic dilation. The CP mapsin the stated example can be taken to have arbitrarily small norm, providingthe first example of a theorem in the classical theory of isometric dilations thatcannot be generalized to the theory of e-dilations of cp-semigroups.Having convinced the reader that subproduct systems are an interesting andimportant object, we turn in the second part of the paper to take a closer lookat the simplest examples of subproduct systems, that is, subproduct systemsof Hilbert spaces over N. We study certain tuples of operators and operatoralgebras that can be naturally associated with every subproduct system, andexplore the relationship between these objects and the subproduct systems thatgive rise to them.

Some preliminaries. M and N will denote von Neumann subalgebras ofB(H), where H is some Hilbert space.In Sections 1 through 5, S will denote a sub-semigroup of Rk+. In fact, in largeparts of the paper S can be taken to be any semigroup with unit, or at least anyOre semigroup (see [25] for a definition), but we prefer to avoid this distraction.

Definition 0.3. A cp-semigroup is a semigroup of CP maps, that is, a familyΘ = {Θs}s∈S of completely positive, contractive and normal maps on M suchthat

Θs+t(a) = Θs(Θt(a)) , s, t ∈ S, a ∈ Mand

Θ0(a) = a , a ∈M.A cp0-semigroup is a semigroup of unital CP maps. An e-semigroup is asemigroup of ∗-endomorphisms. An e0-semigroup is a semigroup of unital ∗-endomorphisms.



For concreteness, one should think of the case S = Nk, where a cp-semigroup isa k-tuple of commuting CP maps, or the case S = Rk+, where a cp-semigroup isa k-parameter semigroup of CP maps, or k mutually commuting one-parametercp-semigroups.

Definition 0.4. Let Θ = {Θs}s∈S be a cp-semigroup acting on a von Neumannalgebra M ⊆ B(H). An e-dilation of Θ is a triple (α,K,R) consisting ofa Hilbert space K ⊇ H (with orthogonal projection PH : K → H), a vonNeumann algebra R ⊆ B(K) that containsM as a corner M = PHRPH , andan e-semigroup α = {αs}s∈S on R such that for all T ∈ R, s ∈ S,

Θs(PHTPH) = PHαs(T )PH .

Definition 0.5. Let A be a C∗-algebra. A Hilbert C∗-correspondences overA is a (right) Hilbert A-module E which carries a non-degenerate, adjointable,left action of A.Definition 0.6. Let M be a W ∗-algebra. A Hilbert W ∗-correspondence overM is a self-adjoint Hilbert C∗-correspondence E over M, such that the mapM→ L(E) which gives rise to the left action is normal.Definition 0.7. Let E be a C∗-correspondence over A, and let H be a Hilbertspace. A pair (σ, T ) is called a completely contractive covariant representationof E on H (or, for brevity, a c.c. representation) if

(1) T : E → B(H) is a completely contractive linear map;(2) σ : A→ B(H) is a nondegenerate ∗-homomorphism; and(3) T (xa) = T (x)σ(a) and T (a ·x) = σ(a)T (x) for all x ∈ E and all a ∈ A.

If A is a W ∗-algebra and E is W ∗-correspondence then we also require that σbe normal.

Given a C∗-correspondence E and a c.c. representation (σ, T ) of E on H ,one can form the Hilbert space E ⊗σ H , which is defined as the Hausdorffcompletion of the algebraic tensor product with respect to the inner product

〈x⊗ h, y ⊗ g〉 = 〈h, σ(〈x, y〉)g〉.One then defines T̃ : E ⊗σ H → H by

T̃ (x⊗ h) = T (x)h.Definition 0.8. A c.c. representation (σ, T ) is called isometric if for all x, y ∈E,

T (x)∗T (y) = σ(〈x, y〉).(This is the case if and only if T̃ is an isometry). It is called fully coisometric

if T̃ is a coisometry.

Given two Hilbert C∗-correspondences E and F over A, the balanced (or inner)tensor product E ⊗F is a Hilbert C∗-correspondence over A defined to be theHausdorff completion of the algebraic tensor product with respect to the innerproduct

〈x⊗ y, w ⊗ z〉 = 〈y, 〈x,w〉 · z〉 , x, w ∈ E, y, z ∈ F.



The left and right actions are defined as a · (x⊗ y) = (a ·x)⊗ y and (x⊗ y)a =x⊗(ya), respectively, for all a ∈ A, x ∈ E, y ∈ F . When working in the contextof W ∗-correspondences, that is, if E and F are W*-correspondences and A isa W ∗-algebra, then E ⊗ F is understood do be the self-dual extension of theabove construction.

Detailed overview of the paper. Subproduct systems, their represen-tations, and their units, are defined in the next section. The following twosections, 2 and 3, can be viewed as a reorganization and sharpening of someknown results, including several new observations.Section 2 establishes the correspondence between cp-semigroups and subprod-uct systems. It is shown that given a subproduct system X of N - correspon-dences and a subproduct system representationR of X on H , we may constructa cp-semigroup Θ acting on N ′. We denote this assignment as Θ = Σ(X,R).Conversely, it is shown that given a cp-semigroup Θ acting on M, there is asubproduct system E (called the Arveson-Stinespring subproduct system of Θ)of M′-correspondences and an injective representation T of E on H such thatΘ = Σ(E, T ). Denoting this assignment as (E, T ) = Ξ(Θ), we have that Σ◦Ξ isthe identity. In Theorem 2.6 we show that Ξ◦Σ is also, after restricting to pairs(X,R) with R an injective representation (and up to some “isomorphism”), theidentity. This allows us to deduce (Corollary 2.8) that a subproduct systemthat is not a product system has no isometric representations. We introducethe Fock spaces associated to a subproduct system and the canonical shift rep-resentations. These constructs allow us to show that every subproduct systemis the Arveson-Stinespring subproduct system of some cp-semigroup.In Section 3 we briefly sketch the picture that is dual to that of Section 2. It isshown that given a subproduct system and a unit of that subproduct systemone may construct a cp-semigroup, and that every cp-semigroup arises this way.In Section 4, we construct for every subproduct system X and every fullycoisometric subproduct system representation T of X on a Hilbert space, asemigroup T̂ of contractions on a Hilbert space that captures “all the informa-tion” about X and T . This construction is a modification of the constructionintroduced in [41] for the case where X is a product system. It turns out that

when X is merely a subproduct system, it is hard to apply T̂ to obtain newresults about the representation T . However, when X is a true product sys-tem T̂ is very handy, and we use it to prove that every e0-semigroup has a∗-automorphic dilation (in a certain sense).Section 5 begins with some general remarks regarding dilations and pieces ofsubproduct system representations, and then the connection between the di-lation theories of cp-semigroups and of representations of subproduct systemsis made. We define the notion of a subproduct subsystem and then we definedilations and pieces of subproduct system representations. These notions gen-eralize the notions of commuting piece or q-commuting piece of [12] and [19],and also generalizes the definition of dilation of a product system representa-tion of [29]. Proposition 5.8, Theorem 5.12 and Corollary 5.13 show that the



1-1 correspondences Σ and Ξ between cp-semigroups and subproduct systemswith representations take isometric dilations of representations to e-dilationsand vice-versa. This is used to obtain an example of three commuting, unitaland contractive CP maps on B(H) for which there exists no e-dilation act-ing on a B(K), and no minimal dilation acting on any von Neumann algebra(Theorem 5.14).In Section 5 we also present a reduction of both the problem of constructing ane0-dilation to a cp0-semigroup, and the problem of constructing an e-dilationto a k-tuple of commuting CP maps with small enough norm, to the problemof embedding a subproduct system in a larger product system. We show thatnot every subproduct system can be embedded in a product system (Propo-sition 5.15), and we use this to construct an example of three commuting CPmaps θ1, θ2, θ3 such that for any λ > 0 the three-tuple λθ1, λθ2, λθ3 has noe-dilation (Theorem 5.16). This unexpected phenomenon has no counterpartin the classical theory of isometric dilations, and provides the first example ofa theorem in classical dilation theory that cannot be generalized to the theoryof e-dilations of cp-semigroups.The developments described in the first part of the paper indicate that sub-product systems are worthwhile objects of study, but to make progress we mustlook at plenty of concrete examples. In the second part of the paper we be-gin studying subproduct systems of Hilbert spaces over the semigroup N. InSection 6 we show that every subproduct system (of W∗-correspondences) overN is isomorphic to a standard subproduct system, that is, it is a subproductsubsystem of the full product system {E⊗n}n∈N for some W∗-correspondenceE. Using the results of the previous section, this gives a new proof to thediscrete analogue of Bhat’s Theorem: every cp0-semigroup over N has an e0-dilation. Given a subproduct system we define the standard X-shift, and weshow that if X is a subproduct subsystem of Y , then the standard X-shift isthe maximal X-piece of the standard Y -shift, generalizing and unifying resultsfrom [12, 19, 39].In Section 7 we explain why subproduct systems are convenient for studyingnoncommutative projective algebraic geometry. We show that every homo-geneous ideal I in the algebra C〈x1, . . . , xd〉 of noncommutative polynomialscorresponds to a unique subproduct system XI , and vice-versa. The represen-tations of XI on a Hilbert space H are precisely determined by the d-tuples inthe zero set of I,

Z(I) = {T = (T1, . . . , Td) ∈ B(H)d : ∀p ∈ I.p(T ) = 0}.A noncommutative version of the Nullstellensatz is obtained, stating that

{p ∈ C〈x1, . . . , xd〉 : ∀T ∈ Z(I).p(T ) = 0} = I.Section 8 starts with a review of a powerful tool, Gelu Popescu’s “PoissonTransform” [38]. Using this tool we derive some basic results (obtained previ-ously by Popescu in [39]) which allow us to identify the operator algebra AXgenerated by the X-shift as the universal unital operator algebra generated by a



row contraction subject to homogeneous polynomial identities. We then provethat every completely bounded representation of a subproduct system X is apiece of a scaled inflation of the X-shift, and derive a related “von Neumanninequality”.In Section 9 we discuss the relationship between a subproduct system X andAX , the (non-selfadjoint operator algebra generated by the X-shift). The mainresult in this section is Theorem 9.7, which says that X ∼= Y if and only if AXis completely isometrically isomorphic to AY by an isomorphism that pre-serves the vacuum state. This result is used in Section 10, where we study theuniversal norm closed unital operator algebra generated by a row contraction(T1, . . . , Td) satisfying the relations

TiTj = qijTjTi , 1 ≤ i < j ≤ d,where q = (qi,j)

di,j=1 ∈ Mn(C) is a matrix such that qj,i = q−1i,j . These non-

selfadjoint analogues of the noncommutative tori, are shown to be classified bytheir subproduct systems when qi,j 6= 1 for all i, j. In particular, when d = 2,we obtain the universal algebra for the relation

T1T2 = qT2T1,

which we call Aq. It is shown that Aq is isomorphically isomorphic to Ar ifand only if q = r or q = r−1.In Section 11 we describe all standard maximal subproduct systems X withdimX(1) = 2 and dimX(2) = 3, and classify their algebras up to isometricisomorphisms.In the closing section of this paper, Section 12, we find that subproduct systemsare also closely related to subshifts and to the subshift C∗-algebras introducedby K. Matsumoto [28]. We show how every subshift gives rise to a subproductsystem, and characterize the subproduct systems that come from subshifts.We use this connection together with the results of Section 8 to describe allrepresentations of subshift C∗-algebras that come from a subshift of finite type(Theorem 12.7).

Acknowledgments. The authors owe their thanks to Eliahu Levy for point-ing out a mistake in a previous version of the paper, and to Michael Skeide, forseveral helpful remarks.



Part 1. Subproduct systems and cp-semigroups

1. Subproduct systems of Hilbert W ∗-correspondences

Definition 1.1. Let N be a von Neumann algebra. A subproduct system ofHilbert W ∗-correspondences over N is a family X = {X(s)}s∈S of HilbertW ∗-correspondences over N such that

(1) X(0) = N ,(2) For every s, t ∈ S there is a coisometric mapping of N -correspondences

Us,t : X(s)⊗X(t)→ X(s+ t),(3) The maps Us,0 and U0,s are given by the left and right actions of N on

X(s),(4) The maps Us,t satisfy the following associativity condition:

(1.1) Us+t,r(Us,t ⊗ IX(r)

)= Us,t+r

(IX(s) ⊗ Ut,r

).

The difference between a subproduct system and a product system is that in asubproduct system the maps Us,t are only required to be coisometric, while in aproduct system these maps are required to be unitaries. Thus, given the imageUs,t(x⊗ y) of x⊗ y in X(s+ t), one cannot recover x and y. Thus, subproductsystems may be thought of as irreversible product systems. The terminologyis, admittedly, a bit awkward. It may be more sensible – however, impossibleat present – to use the term product system for the objects described aboveand to use the term full product system for product system.

Example 1.2. The simplest example of a subproduct system F = FE ={F (n)}n∈N is given by

F (n) = E⊗n,

where E is some W∗-correspondence. F is actually a product system. We shallcall this subproduct system the full product system (over E).

Example 1.3. Let E be a fixed Hilbert space. We define a subproduct system(of Hilbert spaces) SSP = SSPE over N using the familiar symmetric tensorproducts (one can obtain a subproduct system from the anti-symmetric tensorproducts as well). Define

E⊗n = E ⊗ · · · ⊗E,(n times). Let pn be the projection of E

⊗n onto the symmetric subspace ofE⊗n, given by

pnk1 ⊗ · · · ⊗ kn =1

n!

∑

σ∈Sn

kσ−1(1) ⊗ · · · ⊗ kσ−1(n).

We defineSSP (n) = Esn := pnE

⊗n,

the symmetric tensor product of E with itself n times (SSP (0) = C). Wedefine a map Um,n : SSP (m)⊗ SSP (n)→ SSP (m+ n) by

Um,n(x⊗ y) = pm+n(x⊗ y).



The U ’s are coisometric maps because every projection, when considered as amap from its domain onto its range, is coisometric. A straightforward calcula-tion shows that (1.1) holds (see [35, Corollary 17.2]). In these notes we shallrefer to SSP (or SSPE to be precise) as the symmetric subproduct system (overE).

Definition 1.4. Let X and Y be two subproduct systems over the same semi-group S (with families of coisometries {UXs,t}s,t∈S and {UYs,t}s,t∈S). A familyV = {Vs}s∈S of maps Vs : X(s) → Y (s) is called a morphism of subproductsystems if V0 : X(0)→ Y (0) is the identity, if for all s ∈ S \ {0} the map Vs isa coisometric correspondence map, and if for all s, t ∈ S the following identityholds:

(1.2) Vs+t ◦ UXs,t = UYs,t ◦ (Vs ⊗ Vt).V is said to be an isomorphism if Vs is a unitary for all s ∈ S \ {0}. X is saidto be isomorphic to Y if there exists an isomorphism V : X → Y .There is an obvious extension of the above definition to the case where X(0)and Y (0) are ∗-isomorphic. The above notion of morphism is not optimizedin any way. It is simply precisely what we need in order to develop dilationtheory for cp-semigroups.

Definition 1.5. Let N be a von Neumann algebra, let H be a Hilbert space, andlet X be a subproduct system of Hilbert N -correspondences over the semigroupS. Assume that T : X → B(H), and write Ts for the restriction of T to X(s),s ∈ S, and σ for T0. T (or (σ, T )) is said to be a completely contractivecovariant representation of X if

(1) For each s ∈ S, (σ, Ts) is a c.c. representation of X(s); and(2) Ts+t(Us,t(x ⊗ y)) = Ts(x)Tt(y) for all s, t ∈ S and all x ∈ X(s), y ∈

X(t).

T is said to be an isometric (fully coisometric) representation if it is an iso-metric (fully coisometric) representation on every fiber X(s).

Since we shall not be concerned with any other kind of representation, we shallcall a completely contractive covariant representation of a subproduct systemsimply a representation.

Remark 1.6. Item 2 in the above definition of product system can be rewrittenas follows:

T̃s+t(Us,t ⊗ IH) = T̃s(IX(s) ⊗ T̃t).Here T̃s : X(s)⊗σ H → H is the map given by

T̃s(x⊗ h) = Ts(x)h.Example 1.7. We now define a representation T of the symmetric subproductsystem SSP from Example 1.3 on the symmetric Fock space. Denote by F+the symmetric Fock space

F+ =⊕

n∈N

Esn.



For every n ∈ N, the map Tn : SSP (n) = Esn → B(F+) is defined on them-particle space Esm by putting

Tn(x)y = pn+m(x ⊗ y)for all x ∈ X(n), y ∈ Esm. Then T extends to a representation of the sub-product system SSP on F+ (to see that item 2 of Definition 1.5 is satisfied onemay use again [35, Corollary 17.2]).

Definition 1.8. Let X = {X(s)}s∈S be a subproduct system of N -correspondences over S. A family ξ = {ξs}s∈S is called a unit for X if(1.3) ξs ⊗ ξt = U∗s,tξs+t.A unit ξ = {ξs}s∈S is called unital if 〈ξs, ξs〉 = 1N for all s ∈ S, it is calledcontractive if 〈ξs, ξs〉 ≤ 1N for all s ∈ S, and it is called generating if X(s) isspanned by elements of the form(1.4)Us1+···+sn−1,sn(· · ·Us1+s2,s3(Us1,s2(a1ξs1 ⊗ a2ξs2)⊗ a3ξs3 )⊗ · · · ⊗ anξsnan+1),where s = s1 + s2 + · · ·+ sn.From (1.3) follows the perhaps more natural looking

Us,t(ξs ⊗ ξt) = ξs+t.Example 1.9. A unital unit for the symmetric subproduct system SSP fromExample 1.3 is given by defining ξ0 = 1 and

ξn = v ⊗ v ⊗ · · · ⊗ v︸︷︷︸n times

for n ≥ 1. This unit is generating only if E is one dimensional.

2. Subproduct system representations and cp-semigroups

In this section, following Muhly and Solel’s constructions from [29], we showthat subproduct systems and their representations provide a framework fordealing with cp-semigroups, and allow us to obtain a generalization of theclassical result of Wigner that any strongly continuous one-parameter groupof automorphisms of B(H) is given by X 7→ UtXU∗t for some one-parameterunitary group {Ut}t∈R.

2.1. All cp-semigroups come from subproduct system representa-tions.

Proposition 2.1. Let N be a von Neumann algebra and let X be a subproductsystem of N -correspondences over S, and let R be completely contractive co-variant representation of X on a Hilbert space H, such that R0 is unital. Thenthe family of maps

(2.1) Θs : a 7→ R̃s(IX(s) ⊗ a)R̃∗s , a ∈ R0(N )′,



is a semigroup of CP maps on R0(N )′. Moreover, if R is an isometric (a fullycoisometric) representation, then Θs is a ∗-endomorphism (a unital map) forall s ∈ S.Proof. By Proposition 2.21 in [29], {Θs}s∈S is a family of contractive, normal,completely positive maps on R0(N )′. Moreover, these maps are unital if R isa fully coisometric representation, and they are ∗-endomorphisms if R is anisometric representation. It remains is to check that Θ = {Θs}s∈S satisfies thesemigroup condition Θs ◦Θt = Θs+t. Fix a ∈ R0(N )′. For all s, t ∈ S,

Θs(Θt(a)) = R̃s

(IX(s) ⊗

(R̃t(IX(t) ⊗ a)R̃∗t

))R̃∗s

= R̃s(IX(s) ⊗ R̃t)(IX(s) ⊗ IX(t) ⊗ a)(IX(s) ⊗ R̃∗t )R̃∗s= R̃s+t(Us,t ⊗ IG)(IX(s) ⊗ IX(t) ⊗ a)(U∗s,t ⊗ IG)R̃∗s+t= R̃s+t(IX(s·t) ⊗ a)R̃∗s+t= Θs+t(a).

Using the fact that R0 is unital, we have

Θ0(a)h = R̃0(IN ⊗ a)R̃0∗h

= R̃0(IN ⊗ a)(1N ⊗ h)= R0(1N )ah

= ah,

thus Θ0(a) = a for all a ∈ N . �

We will now show that every cp-semigroup is given by a subproduct represen-tation as in (2.1) above. We recall some constructions from [29] (building onthe foundations set in [4]).Fix a CP map Θ on von Neumann algebra M ⊆ B(H). We define M⊗Θ Hto be the Hausdorff completion of the algebraic tensor product M⊗ H withrespect to the sesquilinear positive semidefinite form

〈T1 ⊗ h1, T2 ⊗ h2〉 = 〈h1,Θ(T ∗1 T2)h2〉 .We define a representation πΘ of M on M⊗Θ H by

πΘ(S)(T ⊗ h) = ST ⊗ h,and we define a (contractive) linear map WΘ : H →M⊗H by

WΘ(h) = I ⊗ h.If Θ is unital then WΘ is an isometry, and if Θ is an endomorphism then WΘis a coisometry. The adjoint of WΘ is given by

W ∗Θ(T ⊗ h) = Θ(T )h.For a given CP semigroup Θ on M, Muhly and Solel defined in [29] a W ∗-correspondence EΘ overM′ and a c.c. representation (σ, TΘ) of EΘ on H such



that for all a ∈ M(2.2) Θ(a) = T̃Θ (IEΘ ⊗ a) T̃ ∗Θ.The W ∗-correspondence EΘ is defined as the intertwining space

EΘ = LM(H,M⊗Θ H),where

LM(H,M⊗Θ H) := {X ∈ B(H,M⊗Θ H)∣∣ XT = πΘ(T )X, T ∈M}.

The left and right actions of M′ are given byS ·X = (I ⊗ S)X , X · S = XS

for all X ∈ EΘ and S ∈ M′. The M′-valued inner product on EΘ is de-fined by 〈X,Y 〉 = X∗Y . EΘ is called the Arveson-Stinespring correspondence(associated with Θ).The representation (σ, TΘ) is defined by letting σ = idM′ , the identity repre-sentation of M′ on H , and by defining

TΘ(X) = W∗ΘX.

(idM′ , TΘ) is called the identity representation (associated with Θ). We remarkthat the paper [29] focused on unital CP maps, but the results we cite are truefor nonunital CP maps, with the proofs unchanged.The case where M = B(H) in the following theorem appears, in essence atleast, in [4].

Theorem 2.2. Let Θ = {Θs}s∈S be a cp-semigroup on a von Neumann algebraM ⊆ B(H), and for all s ∈ S let E(s) := EΘs be the Arveson-Stinespringcorrespondence associated with Θs, and let Ts := TΘs denote the identityrepresentation for Θs. Then E = {E(s)}s∈S is a subproduct system of M′-correspondences, and (idM′ , T ) is a representation of E on H that satisfies

(2.3) Θs(a) = T̃s(IE(s) ⊗ a

)T̃ ∗s

for all a ∈ M and all s ∈ S. Ts is injective for all s ∈ S. If Θ is an e-semigroup(cp0-semigroup), then (idM′ , T ) is isometric (fully coisometric).

Proof. We begin by defining the subproduct system maps Us,t : E(s)⊗E(t)→E(s + t). We use the constructions made in [29, Proposition 2.12] and thesurrounding discussion. We define

Us,t = V∗s,tΨs,t ,

where the map

Ψs,t : LM(H,M⊗Θs H)⊗ LM(H,M⊗Θt H)→ LM(H,M⊗Θt M⊗Θs H)is given by Ψs,t(X ⊗ Y ) = (I ⊗X)Y , and

Vs,t : LM(H,M⊗Θs+t H)→ LM(H,M⊗Θt M⊗Θs H)is given by Vs,t(X) = Γs,t ◦X , where Γs,t :M⊗Θs+t H →M⊗ΘtM⊗Θs H isthe isometry

Γs,t : S ⊗Θs+t h 7→ S ⊗Θt I ⊗Θs h.



By [29, Proposition 2.12], Us,t is a coisometric bimodule map for all s, t ∈ S.To see that the U ’s compose associatively as in (1.1), take s, t, u ∈ S, X ∈E(s), Y ∈ E(t), Z ∈ E(u), and compute:

Us,t+u(IE(s) ⊗ Ut,u)(X ⊗ Y ⊗ Z) = Us,t+u(X ⊗ V ∗t,u(I ⊗ Y )Z)= V ∗s,t+u

((I ⊗X)V ∗t,u(I ⊗ Y )Z

)

= Γ∗s,t+u(I ⊗X)Γ∗t,u(I ⊗ Y )Zand

Us+t,u(Us,t ⊗ IE(u))(X ⊗ Y ⊗ Z) = Us+t,u(V ∗s,t(I ⊗X)Y ⊗ Z)= V ∗s+t,u

((I ⊗ V ∗s,t(I ⊗X)Y )Z

)

= Γ∗s+t,u(I ⊗ Γ∗s,t)(I ⊗ I ⊗X)(I ⊗ Y )Z .So it suffices to show that

Γ∗s,t+u(I ⊗X)Γ∗t,u = Γ∗s+t,u(I ⊗ Γ∗s,t)(I ⊗ I ⊗X)It is easier to show that their adjoints are equal. Let a⊗h be a typical elementof M⊗Θs+t+u h.

Γt,u(I ⊗X∗)Γs,t+u(a⊗Θs+t+u h) = Γt,u(I ⊗X∗)(a⊗Θt+u I ⊗Θs h)= Γt,u(a⊗Θt+u X∗(I ⊗Θs h))= a⊗Θu I ⊗Θt X∗(I ⊗Θs h).

On the other hand

(I ⊗ I ⊗X∗)(I ⊗ Γs,t)Γs+t,u(a⊗Θs+t+u h) == (I ⊗ I ⊗X∗)(I ⊗ Γs,t)(a⊗Θu I ⊗Θs+t h)= (I ⊗ I ⊗X∗)(a⊗Θu I ⊗Θt I ⊗Θs h)= a⊗Θu I ⊗Θt X∗(I ⊗Θs h).

This shows that the maps {Us,t} make E into a subproduct system.Let us now verify that T is a representation of subproduct systems. That(idM′ , Ts) is a c.c. representation of E(s) is explained in [29, page 878]. LetX ∈ E(s), Y ∈ E(t).

Ts+t(Us,t(X ⊗ Y )) = W ∗Θs+tΓ∗s,t(I ⊗X)Y,while

Ts(X)Tt(Y ) = W∗ΘsXW

∗ΘtY.

But for all h ∈ H ,WΘtX

∗WΘsh = WΘtX∗(I ⊗Θs h)

= I ⊗Θt X∗(I ⊗Θs h)= (I ⊗X∗)(I ⊗Θt I ⊗Θs h)= (I ⊗X∗)Γs,t(I ⊗Θs+t h)= (I ⊗X∗)Γs,tWΘs+th,



which implies W ∗ΘsXW∗ΘtY = W ∗Θs+tΓ

∗s,t(I ⊗ X)Y , so we have the desired

equality

Ts+t(Us,t(X ⊗ Y )) = Ts(X)Tt(Y ).Equation (2.3) is a consequence of (2.2). The injectivity of the identity repre-sentation has already been noted by Solel in [48] (for all h, g ∈ H and a ∈ M ,〈W ∗ΘXa∗h, g〉 = 〈Xa∗h, I ⊗ g〉 = 〈(I ⊗ a∗)Xh, I ⊗ g〉 = 〈Xh, a⊗ g〉 ). The finalassertion of the theorem is trivial (if Θs is a ∗-endomorphism, then WΘs is acoisometry). �

Definition 2.3. Given a cp-semigroup Θ on a W ∗ algebraM, the pair (E, T )constructed in Theorem 2.2 is called the identity representation of Θ, and Eis called the Arveson-Stinespring subproduct system associated with Θ.

Remark 2.4. If follows from [29, Proposition 2.14], if Θ is an e-semigroup,then the identity representation subproduct system is, in fact, a (full) productsystem.

Remark 2.5. In [27], Daniel Markiewicz has studied the Arveson-Stinespringsubproduct system of a CP0-semigroup over R+ acting on B(H), and has alsoshown that it carries a structure of a measurable Hilbert bundle.

2.2. Essentially all injective subproduct system representationscome from cp-semigroups. The following generalizes and is motivated by[48, Proposition 5.7]. We shall also repeat some arguments from [32, Theorem2.1].By Theorem 2.2, with every cp-semigroup Θ = {Θs}s∈S on M ⊆ B(H) wecan associate a pair (E, T ) - the identity representation of Θ - consisting of asubproduct system E (of correspondences overM′) and an injective subproductsystem c.c. representation T . Let us write (E, T ) = Ξ(Θ). Conversely, given apair (X,R) consisting of a subproduct system X of correspondences over M′and a c.c. representationR of X such that R0 = id, one may define by equation(2.1) a cp-semigroup Θ acting on M, which we denote as Θ = Σ(X,R). Themeaning of equation (2.3) is that Σ ◦ Ξ is the identity map on the set of cp-semigroups of M. We will show below that Ξ ◦ Σ, when restricted to pairs(X,R) such that R is injective, is also, essentially, the identity. When (X,R)is not injective, we will show that Ξ ◦Σ(X,R) “sits inside” (X,R).

Theorem 2.6. Let N be a W∗-algebra, let X = {X(s)}s∈S be a subproductsystem of N -correspondences, and let R be a c.c. representation of X onH, such that σ := R0 is faithful and nondegenerate. Let M ⊆ B(H) be thecommutant σ(N )′ of σ(N ). Let Θ = Σ(X,R), and let (E, T ) = Ξ(Θ). Thenthere is a morphism of subproduct systems W : X → E such that(2.4) Rs = Ts ◦Ws , s ∈ S.W ∗s Ws = IX(s)− qs, where qs is the orthogonal projection of X(s) onto KerRs.In particular, W is an isomorphism if and only if Rs is injective for all s ∈ S.



Remark 2.7. The construction of the morphism W below basically comes from[48, 32], and it remains only to show that it respects the subproduct systemstructure. Some of the details in the proof will be left out.

Proof. We may construct a subproduct system X ′ of M′-correspondences (re-call thatM′ = σ(N )), and a representation R′ of X ′ on H such that R′0 is theidentity, in such a way that (X,R) may be naturally identified with (X ′, R′).Indeed, put

X ′(0) =M′ , X ′(s) = X(s) for s 6= 0,where the inner product is defined by

〈x, y〉X′ = σ(〈x, y〉X),and the left and right actions are defined by

a · x · b := σ−1(a)xσ−1(b),for a, b ∈ M′ and x, y ∈ X ′(s), s ∈ S \ {0}. Defining R′0 = id and W0 = σ;and R′s = Rs for and Ws = id for s ∈ S \ {0}, we have that W is a morphismX → X ′ that sends R to R′.Assume, therefore, that N =M′ and that σ is the identity representation.We begin by defining for every s 6= 0

vs :M⊗Θs H → X(s)⊗Hby

vs(a⊗ h) = (IX(s) ⊗ a)R̃∗sh.It is straightforward to show that for all s ∈ S the map vs is a well-definedisometry. [(IX(s) ⊗M)R̃∗sH ] is invariant under IX(s) ⊗M, thus the projectiononto the orthocomplement of this subspace is in (IX(s)⊗M)′ = L(X(s))⊗ IH ,so it has the form qs ⊗ IH for some projection qs ∈ L(X(s)). In fact, it is easyto check that qs is the orthogonal projection of X(s) onto KerRs.By the definition of vs and by the covariance properties of T , we have for alla ∈ M and b ∈M′,

vs(a⊗ I) = (I ⊗ a)vs , vs(I ⊗ b) = (b ⊗ I)vs.Fix s ∈ S and x ∈ E(s). For all ξ ∈ X(s), h ∈ H , write

ψ(ξ)h = x∗v∗s (ξ ⊗ h).It is easy to verify that the linear map ξ 7→ ψ(ξ) maps X(s) into M′ and isa bounded right module map. From the self duality of X(s) it follows thatthere is a unique element in X(s), which we denote by Vs(x), such that for allξ ∈ X(s), h ∈ H ,(2.5) 〈Vs(x), ξ〉h = x∗v∗s (ξ ⊗ h).The map Vs is then a linear bimodule map. To show that Vs preserves innerproducts, write Lξ, ξ ∈ X(s), for the operator Lξ : H → X(s)⊗H that mapsh to ξ ⊗ h and note that equation (2.5) becomes

L∗Vs(x)Lξ = x∗v∗sLξ , ξ ∈ X(s),



or LVs(x) = vsx, for all x ∈ E(s). But since vs preserves inner products, wehave for all x, y ∈ E(s):

〈x, y〉 = x∗y = x∗v∗svsy = L∗Vs(x)LVs(y) = 〈Vs(x), Vs(y)〉.

We now prove that VsV∗s = IX(s)−qs. ξ ∈ ImV ⊥s if and only if L∗ξvsE(s)H = 0.

But by [29, Lemma 2.10], E(s)H =M⊗Θs H , thus L∗ξvsE(s)H = 0 if and onlyif 〈ξ, η〉 = 0 for all η ∈ (IX(s) − qs)X(s), which is the same as ξ ∈ qsX(s).Fix h, k ∈ H . For x ∈ E(s), we compute:

〈Ts(x)h, k〉 = 〈W ∗Θsxh, k〉= 〈xh, I ⊗Θs k〉= 〈vsxh, vs(I ⊗Θs k)〉= 〈Vs(x)⊗ h, R̃∗sk〉= 〈Rs(Vs(x))h, k〉,

thus Ts = Rs ◦ Vs for all s ∈ S. Define Ws = V ∗s . Then Ts = Rs ◦ W ∗s .Multiplying both sides by Ws we obtain Ts ◦Ws = Rs ◦W ∗s Ws. But W ∗s Ws =I − qs is the orthogonal projection onto (KerRs)⊥, thus we obtain (2.4).Finally, we need to show that W = {Ws} respects the subproduct systemstructure: for all s, t ∈ S, x ∈ X(s) and y ∈ X(t), we must show that

Ws+t(UXs,t(x⊗ y)) = UEs,t(Ws(x) ⊗Wt(y)).

Since Ts+t is injective, it is enough to show that after applying Ts+t to bothsides of the above equation we get the same thing. But Ts+t applied to the lefthand side gives

Ts+tWs+t(UXs,t(x⊗ y)) = Rs+t(UXs,t(x⊗ y)) = Rs(x)Rt(y),

and Ts+t applied to the right hand side gives

Ts+t(UEs,t(Ws(x) ⊗Wt(y))) = Ts(Ws(x))Tt(Wt(y)) = Rs(x)Rt(y).

�

Corollary 2.8. Let X be a subproduct system that has an isometric represen-tation V such that V0 is faithful and nondegenerate. Then X is a (full) productsystem.

Proof. Let Θ = Σ(X,V ). Then Θ is an e-semigroup. Thus, if (E, T ) = Ξ(Θ)is the identity representation of Θ, then, by Remark 2.4, E is a (full) productsystem. But if V0 is faithful and V is isometric then V is injective. By theabove theorem, X is isomorphic to E, so it is a product system. �

2.3. Subproduct systems arise from cp-semigroups. The shift rep-resentation. A question arises: does every subproduct system arise as theArveson-Stinespring subproduct system associated with a cp-semigroup? ByTheorem 2.6, this is equivalent to the question does every subproduct system



have an injective representation? We shall answer this question in the affir-mative by constructing for every such subproduct system a canonical injectiverepresentation.The following constructs will be of most interest when S is a countable semi-group, such as Nk.

Definition 2.9. Let X = {X(s)}s∈S be a subproduct system. The X-Fockspace FX is defined as

FX =⊕

s∈S

X(s).

The vector Ω := 1 ∈ N = X(0) is called the vacuum vector of FX . The X-shiftrepresentation of X on FX is the representation

SX : X → B(FX),given by SX(x)y = UXs,t(x⊗ y), for all x ∈ X(s), y ∈ X(t) and all s, t ∈ S.

Strictly speaking, SX as defined above is not a representation because it rep-resents X on a C∗-correspondence rather than on a Hilbert space. However,since for any C∗-correspondence E, L(E) is a C∗-algebra, one can compose afaithful representation π : L(E) → B(H) with SX to obtain a representationon a Hilbert space.

A direct computation shows that S̃Xs : X(s)⊗ FX → FX is a contraction, andalso that SX(x)SX(y) = SX(UXs,t(x ⊗ y)) so SX is a completely contractiverepresentation of X . SX is also injective because SX(x)Ω = x for all x ∈ X .Thus,

Corollary 2.10. Every subproduct system is the Arveson-Stinespring sub-product system of a cp-semigroup.

3. Subproduct system units and cp-semigroups

In this section, following Bhat and Skeide’s constructions from [15], we showthat subproduct systems and their units may also serve as a tool for studyingcp-semigroups.

Proposition 3.1. Let N be a von Neumann algebra and let X be a subproductsystem of N -correspondences over S, and let ξ = {ξs}s∈S be a contractive unitof X, such that ξ0 = 1N . Then the family of maps

(3.1) Θs : a 7→ 〈ξs, aξs〉 ,is a semigroup of CP maps on N . Moreover, if ξ is unital, then Θs is a unitalmap for all s ∈ S.

Proof. It is standard that Θs given by (3.1) is a contractive completely positivemap on N , which is unital if and only if ξ is unital. The fact that Θs is normalgoes a little bit deeper, but is also known (one may use [29, Remark 2.4(i)]).



We show that {Θs}s∈S is a semigroup. It is clear that Θ0(a) = a for all a ∈ N .For all s, t ∈ S,

Θs(Θt(a)) = 〈ξs, 〈ξt, aξt〉 ξs〉= 〈ξt ⊗ ξs, , aξt ⊗ ξs〉=〈U∗t,sξs+t, , aU

∗t,sξs+t

〉

= 〈ξs+t, aξs+t〉= Θs+t(a).

�

We recall a central construction in Bhat and Skeide’s approach to dilation ofcp-semigroup [15], that goes back to Paschke [36]. LetM be a W ∗-algebra, andlet Θ be a normal completely positive map on M 3. The GNS representationof Θ is a pair (FΘ, ξΘ) consisting of a Hilbert W

∗-correspondence FΘ and avector ξΘ ∈ FΘ such that

Θ(a) = 〈ξΘ, aξΘ〉 for all a ∈ M.FΘ is defined to be the correspondence M⊗ΘM - which is the self-dual ex-tension of the Hausdorff completion of the algebraic tensor product M⊗Mwith respect to the inner product

〈a⊗ b, c⊗ d〉 = b∗Θ(a∗c)d.ξΘ is defined to be ξΘ = 1 ⊗ 1. Note that ξΘ is a unit vector, that is -〈ξΘ, ξΘ〉 = 1, if and only if Θ is unital.Theorem 3.2. Let Θ = {Θs}s∈S be a cp-semigroup on a W ∗-algebra M.For every s ∈ S let (F (s), ξs) be the GNS representation of Θs. Then F ={F (s)}s∈S is a subproduct system of M-correspondences, and ξ = {ξs}s∈S isa generating contractive unit for F that gives back Θ by the formula

(3.2) Θs(a) = 〈ξs, aξs〉 for all a ∈M.Θ is a cp0-semigroup if and only if ξ is a unital unit.

Proof. For all s, t ∈ S define a map Vs,t : F (s + t) → F (s) ⊗ F (t) by sendingξs+t to ξs ⊗ ξt and extending to a bimodule map. Because

〈aξs ⊗ ξtb, cξs ⊗ ξtd〉 = 〈ξtb, 〈aξs, cξs〉 ξtd〉= 〈ξtb,Θs(a∗c)ξtd〉= b∗ 〈ξt,Θs(a∗c)ξt〉 d= b∗Θt+s(a

∗c)d

= 〈aξt+sb, cξt+sd〉 ,Vs,t extends to a well defined isometric bimodule map from F (s+t) into F (s)⊗F (t). We define the map Us,t to be the adjoint of Vs,t (here it is important that

3The construction works also for completely positive maps on unital C∗-algebras, but inTheorem 3.2 below we will need to work with normal maps on W∗-algebras.



we are working with W ∗ algebras - in general, an isometry from one HilbertC∗-module into another need not be adjointable, but bounded module mapsbetween self-dual Hilbert modules are always adjointable, [36, Proposition 3.4]).The collection {Us,t}s,t∈S makes F into a subproduct system. Indeed, thesemaps are coisometric by definition, and they compose in an associative manner.To see the latter, we check that (IF (r) ⊗ Vs,t)Vr,s+t = (Vr,s ⊗ IF (t))Vr+s,t andtake adjoints.

(IF (r) ⊗ Vs,t)Vr,s+t(aξr+s+tb) = (IF (r) ⊗ Vs,t)(aξr ⊗ ξs+tb)= aξr ⊗ ξs ⊗ ξtb.

Similarly, (Vr,s ⊗ IF (t))Vr+s,t(aξr+s+tb) = aξr ⊗ ξs ⊗ ξtb. Since F (r + s+ t) isspanned by linear combinations of elements of the form aξr+s+tb, the U ’s makeF into a subproduct system, and ξ is certainly a unit for F . Equation (3.2)follows by definition of the GNS representation. Now,

〈ξs, ξs〉 = Θs(1) , s ∈ S,so ξ is a contractive unit because Θs(1) ≤ 1, and ξ it is unital if and only ifΘs is unital for all s. ξ is in fact more then just a generating unit, as F (s) isspanned by elements with the form described in equation (1.4) with (s1, . . . , sn)a fixed n-tuple such that s1 + · · ·+ sn = s. �

Definition 3.3. Given a cp-semigroup Θ on a W ∗ algebra M, the subproductsystem F and the unit ξ constructed in Theorem 3.2 are called, respectively,the GNS subproduct system and the GNS unit of Θ. The pair (F, ξ) is calledthe GNS representation of Θ.

Remark 3.4. There is a precise relationship between the identity represen-tation (Definition 2.3) and the GNS representation of a cp-semigroup. TheGNS representation of a CP map is the dual of the identity representation ina sense that is briefly described in [31]. This notion of duality has been usedto move from the product-system-and-representation picture to the product-system-with-unit picture, and vice versa. See for example [44] and the ref-erences therein. It is more-or-less straightforward to use this duality to getTheorem 3.2 from Theorem 2.2 (or the other way around).

4. ∗-automorphic dilation of an e0-semigroupWe now apply some of the tools developed above to dilate an e0-semigroup to asemigroup of ∗-automorphisms. We shall need the following proposition, whichis a modification (suited for subproduct systems) of the method introducedin [41] for representing a product system representation as a semigroup ofcontractive operators on a Hilbert space.

Proposition 4.1. Let N be a von Neumann algebra and let X be a subproductsystem of W ∗-correspondences over S. Let (σ, T ) be a fully coisometric covari-ant representation of X on the Hilbert space H, and assume that σ is unital.



Denote

Hs :=(X(s)⊗σ H

)/KerT̃s

and

H =⊕

s∈S

Hs.

Then there exists a semigroup of coisometries T̂ = {T̂s}s∈S on H such that forall s ∈ S, x ∈ X(s) and h ∈ H,

T̂s (δs · x⊗ h) = Ts(x)h.T̂ also has the property that for all s ∈ S and all t ≥ s(4.1) T̂ ∗s T̂s

∣∣Ht

= IHt , (t ≥ s).

Proof. First, we note that the assumptions on σ and on the left action of Nimply that H0 ∼= H via the identification a ⊗ h ↔ σ(a)h. This identificationwill be made repeatedly below.Define H0 to be the space of all finitely supported functions f on S such thatfor all s ∈ S,

f(s) ∈ Hs.H0 is generated by the functions δs ·ξ, where δs is the function taking the value1 at s and 0 elsewhere, and ξ ∈ Hs. We equip H0 with the inner product

〈δs · ξ, δt · η〉 = δs,t〈ξ, η〉,for all s, t ∈ S, ξ ∈ Hs, η ∈ Ht (here δs,t is Kronecker’s delta function). Let Hbe the completion of H0 with respect to this inner product. We have

H ∼=⊕

s∈S

Hs.

It will sometimes be convenient to identify the subspace δs ·Hs ⊆ H with Hs,and for s = 0 this gives us an inclusion H ⊆ H. We define a family T̂ = {T̂s}s∈Sof operators on H0 as follows. First, we define T̂0 to be the identity. Nowassume that s > 0. If t ∈ S and t � s, then we define T̂s(δt · ξ) = 0 for allξ ∈ Ht. If t ≥ s > 0 we would like to define (as we did in [41])

(4.2) T̂s (δt · (xt−s ⊗ xs ⊗ h)) = δt−s ·(xt−s ⊗ T̃s(xs ⊗ h)

),

but since X is not a true product system, we cannot identify X(t− s)⊗X(s)with X(t). For a fixed t > 0, we define for all s ≤ t, ξ ∈ X(t) and h ∈ H

Ťs (δt · (ξ ⊗ h)) = δt−s ·(

(IX(t−s) ⊗ T̃s)(U∗t−s,sξ ⊗ h)).

Ťs can be extended to a well defined contraction from X(t)⊗H to X(t−s)⊗H ,for all t ≥ s, and has an adjoint given by

(4.3) Ť ∗s δt−s · η ⊗ h = δt ·(

(Ut−s,s ⊗ IH)(η ⊗ T̃ ∗s h)).



We are going to obtain T̂s as the map Ht → Ht−s induced by Ťs. Let Y ∈ Htsatisfy T̃t(Y ) = 0. We shall show that Ťsδt · Y = 0 in δt−s ·Ht−s. But

Ťsδt · Y = δt−s ·(

(IX(t−s) ⊗ T̃s)(U∗t−s,s ⊗ IH)Y),

and

T̃t−s

((IX(t−s) ⊗ T̃s)(U∗t−s,s ⊗ IH)Y

)(∗) = T̃t(Ut−s,s ⊗ IH)(U∗t−s,s ⊗ IH)Y

(∗∗) = T̃t(Y ) = 0,where the equation marked by (*) follows from the fact that T is a representa-tion of subproduct systems, and the one marked by (**) follows from the factthat Ut−s,s is a coisometry. Thus, for all s, t ∈ S,

Ťs

(δt ·KerT̃t

)⊆ δt−s ·KerT̃t−s,

thus Ťs induces a well defined contraction T̂s on H given by

(4.4) T̂s (δt · (ξ ⊗ h)) = δt−s ·(

(IX(t−s) ⊗ T̃s)(U∗t−s,sξ ⊗ h)),

where ξ⊗h and (IX(t−s)⊗T̃s)(U∗t−s,sξ⊗h) stand for these elements’ equivalenceclasses in

(X(t) ⊗ H

)/KerT̃t and

(X(t − s) ⊗ H

)/KerT̃t−s, respectively. It

follows that we have the following, more precise, variant of (4.2):

T̂s (δt · (Ut−s,s(xt−s ⊗ xs)⊗ h)) = δt−s ·(xt−s ⊗ T̃s(xs ⊗ h)

).

In particular,

T̂s (δs · xs ⊗ h) = Ts(xs)h,for all s ∈ S, xs ∈ X(s), h ∈ H .It will be very helpful to have a formula for T̂ ∗s as well. Assume that

∑i ξi⊗hi ∈

KerT̃t.

Ť ∗s

(δt ·∑

i

ξi ⊗ hi)

= δs+t ·(

(Ut,s ⊗ IH)(∑

i

ξi ⊗ T̃ ∗s hi)),

and applying T̃s+t to the right hand side (without the δ) we get

T̃s+t

((Ut,s ⊗ IH)(

∑

i

ξi ⊗ T̃ ∗s hi))

= T̃t(IX(t) ⊗ T̃s)(∑

i

ξi ⊗ T̃ ∗s hi)

= T̃t(∑

i

ξi ⊗ T̃sT̃ ∗s hi)

= T̃t(∑

i

ξi ⊗ hi) = 0,

because T is a fully coisometric representation. So

Ť ∗s

(δt ·KerT̃t

)⊆ δs+t ·KerT̃s+t,



and this means that Ť ∗s induces on H a well defined contraction which is equalto T̂ ∗s , and is given by the formula (4.3).

We now show that T̂ is a semigroup. Let s, t, u ∈ S. If either s = 0 or t = 0then it is clear that the semigroup property T̂sT̂t = T̂s+t holds. Assume that

s, t > 0. If u � s+ t, then both T̂sT̂t and T̂s+t annihilate δu · ξ, for all ξ ∈ Hu.Assuming u ≥ s+ t, we shall show that T̂sT̂t and T̂s+t agree on elements of theform

Z = δu ·(Uu−t,t(Uu−t−s,s ⊗ I)(xu−s−t ⊗ xs ⊗ xt)

)⊗ h,

and since the set of all such elements is total in Hu, this will establish thesemigroup property.

T̂sT̂tZ = T̂s

(δu−t

(Uu−t−s,s(xu−s−t ⊗ xs)⊗ T̃t(xt ⊗ h)

))

= δu−s−t

(xu−s−t ⊗ T̃s(xs ⊗ T̃t(xt ⊗ h))

)

= δu−s−t

(xu−s−t ⊗ T̃s(I ⊗ T̃t)(xs ⊗ xt ⊗ h)

)

= δu−s−t

(xu−s−t ⊗ T̃s+t (Us,t(xs ⊗ xt)⊗ h)

)

= T̂t+sδu · (Uu−t−s,t+s (xu−s−t ⊗ Us,t(xs ⊗ xt))⊗ h)= T̂t+sZ.

The final equality follows from the associativity condition (1.1).

To see that T̂ is a semigroup of coisometries, we take ξ ∈ X(t), h ∈ H , andcompute

T̃t

(T̂sT̂

∗s δt · (ξ ⊗ h)

)=

= T̃t

((IX(t) ⊗ T̃s)(U∗t,s ⊗ IH)(Ut,s ⊗ IH)(IX(t) ⊗ T̃ ∗s )(ξ ⊗ h)

)

= T̃s+t(Ut,s ⊗ IH)(IX(t) ⊗ T̃ ∗s )(ξ ⊗ h)= T̃t(ξ ⊗ T̃sT̃ ∗s h) = T̃t(ξ ⊗ h),

so T̂sT̂∗s is the identity on Ht for all t ∈ S, thus T̂sT̂ ∗s = IH. Equation (4.1)

follows by a similar computation, which is omitted. �

We can now obtain a ∗-automorphic dilation for any e0-semigroup over anysubsemigroup of Rk+. The following result should be compared with similar-looking results of Arveson-Kishimoto [8], Laca [25], Skeide [45], and Arveson-Courtney [7] (none of these cited results is strictly stronger or weaker than theresult we obtain for the case of e0-semigroups).

Theorem 4.2. Let Θ be a e0-semigroup acting on a von Neumann algebraM. Then Θ can be dilated to a semigroup of ∗-automorphisms in the follow-ing sense: there is a Hilbert space K, an orthogonal projection p of K onto asubspace H of K, a normal, faithful representation ϕ : M → B(K) such that



ϕ(1) = p, and a semigroup α = {αs}s∈S of ∗-automorphisms on B(K) suchthat for all a ∈ M and all s ∈ S(4.5) αs(ϕ(a))

∣∣H

= ϕ(Θs(a)),

so, in particular,

(4.6) pαs(ϕ(a))p = ϕ(Θs(a)).

The projection p is increasing for α, in the sense that for all s ∈ S,(4.7) αs(p) ≥ p.Remark 4.3. Another way of phrasing the above theorem is by using theterminology of “weak Markov flows”, as used in [15]. Denoting ϕ by j0, anddefining js := αs ◦ j0, we have that (B(K), j) is a weak Markov flow for Θ onK, which just means that for all t ≤ s ∈ S and all a ∈ M,(4.8) jt(1)js(a)jt(1) = jt(Θs−t(a)).

Equation (4.8) for t = 0 is just (4.6), and the case t ≥ 0 follows from the caset = 0.

Remark 4.4. The assumption that Θ is a unital semigroup is essential, since(4.6) and (4.7) imply that Θ(1) = 1.

Remark 4.5. It is impossible, in the generality we are working in, to hope fora semigroup of automorphisms that extends Θ in the sense that

(4.9) αs(ϕ(a)) = ϕ(Θs(a)),

because that would imply that Θ is injective.

Proof. Let (E, T ) be the identity representation of Θ. Since Θ preserves the

unit, T is a fully coisometric representation. Let T̂ and H be the semigroupand Hilbert space representing T as described in Proposition 4.1. {T̂ ∗s }s∈S is acommutative semigroup of isometries. By a theorem of Douglas [20], {T̂ ∗s }s∈Scan be extended to a semigroup {V̂ ∗s }s∈S of unitaries acting on a space K ⊇ H.We obtain a semigroup of unitaries V = {V̂s}s∈S that is a dilation of T̂ , that is

PHV̂s∣∣H

= T̂s , s ∈ S.For any b ∈ B(K), and any s ∈ S, we define

αs(b) = V̂sbV̂∗s .

Clearly, α = {αs}s∈S is a semigroup of ∗-automorphisms.Put p = PH, the orthogonal projection of K onto H. Define ϕ :M→ B(K) byϕ(a) = p(I ⊗ a)p, where I ⊗ a : H → H is given by

(I ⊗ a)δt · x⊗ h = δt · x⊗ ah , x⊗ h ∈ E(t)⊗H.ϕ is well defined because T is an isometric representation (so KerT̃t is alwayszero). We have that ϕ is a faithful, normal ∗-representation (the fact that T0 isthe identity representation ensures that ϕ is faithful). It is clear that ϕ(1) = p.



To see (4.7), we note that since V̂ ∗s is an extension of T̂∗s , we have T̂

∗s = V̂

∗s p =

pV̂ ∗s p, thus

pαs(p)p = pV̂spV̂∗s p

= pV̂sV̂∗s p

= p,

that is, pαs(p)p = p, which implies that αs(p) ≥ p.We now prove (4.6). Let δt · x⊗ h be a typical element of H. We compute

pαs(ϕ(a))pδt · x⊗ h = pV̂sp(I ⊗ a)pV̂ ∗s pδt · x⊗ h= T̂s(I ⊗ a)T̂ ∗s δt · x⊗ h

= T̂s(I ⊗ a)δs+t · (Ut,s ⊗ IH)(x⊗ T̃ ∗s h

)

= T̂sδs+t · (Ut,s ⊗ IH)(x⊗ (I ⊗ a)T̃ ∗s h

)

= δt · x⊗(T̃s(I ⊗ a)T̃ ∗s h

)

= δt · x⊗ (Θs(a)h)= ϕ(Θs(a))δt · x⊗ h.

Since both pαs(ϕ(a))p and ϕ(Θs(a)) annihilate K ⊖H, we have (4.6).To prove (4.5), it just remains to show that

pαs(ϕ(a))∣∣H

= αs(ϕ(a))∣∣H,

that is, that αs(ϕ(a))H ⊆ H. Now, V̂ ∗s is an extension of T̂ ∗s . Moreover (4.1)shows that if ξ ∈ Hu with u ≥ s, then ‖T̂s(ξ)‖ = ‖ξ‖. Thus‖ξ‖2 = ‖V̂sξ‖2 = ‖PHV̂sξ‖2 + ‖(IK − PH)V̂sξ‖2 = ‖T̂sξ‖2 + ‖(IK − PH)V̂sξ‖2.So V̂sξ = T̂sξ for ξ ∈ Hu with u ≥ s. Now, for a typical element δt · x ⊗ h inHt, t ∈ S, we have

αs(ϕ(a))δt · x⊗ h = V̂s(I ⊗ a)V̂ ∗s δt · x⊗ h= V̂s(I ⊗ a)T̂ ∗s δt · x⊗ h

= V̂sδs+t · (Us,t ⊗ IH)(x⊗ (I ⊗ a)T̃ ∗s h

)

= T̂sδs+t · (Us,t ⊗ IH)(x⊗ (I ⊗ a)T̃ ∗s h

)∈ H,

because δs+t · x⊗ (I ⊗ a)T̃ ∗s h ∈ Hs+t, and s+ t ≥ s. �

5. Dilations and pieces of subproduct system representations

5.1. Dilations and pieces of subproduct system representations.

Definition 5.1. Let X and Y be subproduct systems of M correspondences(M a W∗-algebra) over the same semigroup S. Denote by UXs,t and UYs,t thecoisometric maps that make X and Y , respectively, into subproduct systems.



X is said to be a subproduct subsystem of Y (or simply a subsystem of Y forshort) if for all s ∈ S the space X(s) is a closed subspace of Y (s), and if theorthogonal projections ps : Y (s)→ X(s) are bimodule maps that satisfy(5.1) ps+t ◦ UYs,t = UXs,t ◦ (ps ⊗ pt) , s, t ∈ S.One checks that if X is a subproduct subsystem of Y then

(5.2) ps+t+u ◦ UYs,t+u(I ⊗ (pt+u ◦ UYt,u)) = ps+t+u ◦ UYs+t,u((ps+t ◦ UYs,t)⊗ I),for all s, t, u ∈ S. Conversely, given a subproduct system Y and a family oforthogonal projections {ps}s∈S that are bimodule maps satisfying (5.2), thenby defining X(s) = psY (s) and U

Xs,t = ps+t ◦ UYs,t one obtains a subproduct

subsystem X of Y (with (5.1) satisfied).The following proposition is a consequence of the definitions.

Proposition 5.2. There exists a morphism X → Y if and only if Y is iso-morphic to a subproduct subsystem of X.

Remark 5.3. In the notation of Theorem 2.6, we may now say that given a sub-product system X and a representation R of X , then the Arveson-Stinespringsubproduct system E of Θ = Σ(X,R) is isomorphic to a subproduct subsystemof X .

The following definitions are inspired by the work of Bhat, Bhattacharyya andDey [12].

Definition 5.4. Let X and Y be subproduct systems of W∗-correspondences(over the same W∗-algebra M) over S, and let T be a representation of Y ona Hilbert space K. Let H be some fixed Hilbert space, and let S = {Ss}s∈S bea family of maps Ss : X(s)→ B(H). (Y, T,K) is called a dilation of (X,S,H)if

(1) X is a subsystem of Y ,(2) H is a subspace of K, and

(3) for all s ∈ S, T̃ ∗s H ⊆ X(s)⊗H and T̃ ∗s∣∣H

= S̃∗s .

In this case we say that S is an X-piece of T , or simply a piece of T . T is saidto be an isometric dilation of S if T is an isometric representation.

The third item can be replaced by the three conditions

1’ T0(·)PH = PHT0(·)PH = S0(·),2’ PH T̃s

∣∣X(s)⊗H

= S̃s for all s ∈ S, and3’ PH T̃s

∣∣Y (s)⊗K⊖X(s)⊗H

= 0.

So our definition of dilation is identical to Muhly and Solel’s definition of di-lation of representations when X = Y is a product system [29, Theorem andDefinition 3.7].

Proposition 5.5. Let T be a representation of Y , let X be a subproduct sub-system of Y , and let S an X-piece of T . Then S is a representation of X.



Proof. S is a completely contractive linear map as the compression of a com-pletely contractive linear map. Item 1’ above together with the coinvariance ofT imply that S is coinvariant: if a, b ∈ M and x ∈ X(s), then

Ss(axb) = PHTs(axb)PH = PHT0(a)Ts(x)T0(b)PH

= PHT0(a)PHTs(x)PHT0(b)PH

= S0(a)Ss(x)S0(b).

Finally, (using Item 3’ above),

Ss+t(UXs,t(x ⊗ y))h = Ss+t(ps+tUYs,t(x ⊗ y))h

= S̃s+t(ps+tUYs,t(x ⊗ y)⊗ h)

= PH T̃s+t(UYs,t(x⊗ y)⊗ h)

= PHTs(x)Tt(y)h

= PHTs(x)PHTt(y)h

= Ss(x)St(y)h.

�

Example 5.6. Let E be a Hilbert space of dimension d, and let X be thesymmetric subproduct system constructed in Example 1.3. Fix an orthonor-mal basis {e1, . . . , en} of E. There is a one-to-one correspondence betweenc.c. representations S of X (on some H) and commuting row contractions(S1, . . . , Sd) (of operators on H), given by

S ↔ S = (S(e1), . . . , S(ed)).If Y is the full product system over E, then any dilation (Y, T,K) gives rise toa tuple T = (T (e1), . . . , T (ed)) that is a dilation of S in the sense of [12], andvice versa. Moreover, S is then a commuting piece of T in the sense of [12].

Consider a subproduct system Y and a representation T of Y on K. Let X besome subproduct subsystem of Y . Define the following set of subspaces of K:

(5.3) P(X,T ) = {H ⊆ K : T̃ ∗sH ⊆ X(s)⊗H for all s ∈ S}.As in [12], we observe that P(X,T ) is closed under closed linear spans (andintersections), thus we may define

KX(T ) =∨

H∈P(X,T )

H.

KX(T ) is the maximal element of P(X,T ).Definition 5.7. The representation TX of X on KX(T ) given by

TX(x)h = PKX (T )T (x)h,

for x ∈ X(s) and h ∈ KX(T ), is called the maximal X-piece of T .By Proposition 5.5, TX is indeed a representation of X .



5.2. Consequences in dilation theory of cp-semigroups.

Proposition 5.8. Let X and Y be subproduct systems of W∗-correspondences(over the same W∗-algebra M) over S, and let S and T be representations ofX on H and of Y on K, respectively. Assume that (Y, T,K) is a dilation of(X,S,H). Then the cp-semigroup Θ acting on V0(M)′, given by

Θs(a) = T̃s(IY (s) ⊗ a)T̃ ∗s , a ∈ V0(M)′,is a dilation of the cp-semigroup Φ acting on T0(M)′ given by

Φs(a) = S̃s(IX(s) ⊗ a)S̃∗s , a ∈ T0(M)′,

in the sense that for all b ∈ V0(M)′ and all s ∈ S,Φs(PHbPH) = PHΘs(b)PH .

Proof. This follows from the definitions. �

Although the above proposition follows immediately from the definitions, wehope that it will prove to be important in the theory of dilations of cp-semigroups, because it points to a conceptually new way of constructing di-lations of cp-semigroups, as the following proposition and corollary illustrate.

Proposition 5.9. Let X = {X(s)}s∈S be a subproduct system, and let S bea fully coisometric representation of X on H such that S0 is unital. If thereexists a (full) product system Y = {Y (s)}s∈S such that X is a subproductsubsystem of Y , then S has an isometric and fully coisometric dilation.

Proof. Define a representation T of Y on H by

(5.4) Ts = Ss ◦ ps,where, as above, ps is the orthogonal projection Y (s) → X(s). A straightfor-ward verification shows that T is indeed a fully coisometric representation of Yon H . By [43, Theorem 5.2], (Y, T,H) has a minimal isometric and fully coiso-metric dilation (Y, V,K). (Y, V,K) is also clearly a dilation of (X,S,H). �

Corollary 5.10. Let Θ = {Θs}s∈S be a cp0-semigroup and let (E, T ) = Ξ(Θ)be the Arveson-Stinespring representation of Θ. If there is a (full) productsystem Y such that E is a subproduct subsystem of Y , then Θ has an e0-dilation.

Proof. Combine Propositions 2.1, 5.8 and 5.9. �

Thus, the problem of constructing e0-dilations to cp0-semigroups is reduced tothe problem of embedding a subproduct system into a full product system. Inthe next subsection we give an example of a subproduct system that cannotbe embedded into full product system. When this can be done in general is achallenging open question.



Corollary 5.11. Let Θ = {Θs}s∈Nk be a cp-semigroup generated by k com-muting CP maps θ1, . . . , θk, and let (E, T ) = Ξ(Θ) be the Arveson representa-tion of Θ. Assume, in addition, that

k∑

i=1

‖θi‖ ≤ 1.

If there is a (full) product system Y such that E is a subproduct subsystem ofY , then Θ has an e-dilation.

Proof. As in (5.4), we may extend T to a product system representation of Yon H , which we also denote by T . Denote by ei the element of Nk with 1 inthe ith element and zeros elsewhere. Then

k∑

i=1

‖T̃ei T̃ ∗ei‖ =k∑

i=1

‖θi‖ ≤ 1.

By the methods of [41], one may show that S has a minimal (regular) isometricdilation. This isometric dilation provides the required e-dilation of Θ. �

Theorem 5.12. Let M ⊆ B(H) be a von Neumann algebra, let X be a sub-product system of M′-correspondences, and let R be an injective representa-tion of X on a Hilbert space H. Let Θ = Σ(X,R) be the cp-semigroup act-ing on R0(M′)′ given by (2.1). Assume that (α,K,R) is an e-dilation of Θ,and let (Y, V ) = Ξ(α) be the Arveson-Stinespring subproduct system of α to-gether with the identity representation. Assume, in addition, that the mapR′ ∋ b 7→ PHbPH is a ∗-isomorphism of R′ onto R0(M′). Then (Y, V,K) is adilation of (X,R,H).

Proof. For every s ∈ S, define Ws : Y (s) → B(H) by Ws(y) = PHVs(y)PH .We claim that W = {Ws}s∈S is a representation of Y on H . First, notethat PHαs(I − PH)PH = Θs(PH(I − PH)PH) = 0, thus PH Ṽs(I ⊗ (I −PH))Ṽ

∗s PH = 0, and consequently PH Ṽs(I ⊗ PH) = PH Ṽs. It follows that

Ws(y) = PHVs(y)PH = PHVs(y). From this it follows that

Ws(y1)Wt(y2) = PHVs(y1)PHVt(y2) = PHVs(y1)Vt(y2)

= PHVs+t(UYs,t(y1 ⊗ y2)) = Ws+t(UYs,t(y1 ⊗ y2)).

By Theorem 2.6, we may assume that (X,R) = (E, T ) = Ξ(Θ) is the Arveson-Stinespring representation of Θ. Because α is a dilation of Θ, we have

W̃s(I ⊗ a)W̃ ∗s = PH Ṽs(I ⊗ a)Ṽ ∗s PH = Θs(a),That is, Θ = Σ(Y,W ). Thus, by Theorem 2.6 and Remark 5.3, we may assumethat E is a subproduct subsystem of Y , and that Ts ◦ ps = Ws, ps being theprojection of Y (s) onto E(s). In other words, for all y ∈ Y ,

T̃s(ps ⊗ IH) = PHW̃s.



Therefore, W̃ ∗s H ⊆ E(s)⊗H , and W̃ ∗s∣∣H

= T̃ ∗s . That is, (Y,W,H) is a dilation

of (E, T,H). But (Y, V,K) is a dilation of (Y,W,H), so it is also a dilation of(E, T,H). �

The assumption that R′ ∋ b 7→ PHbPH ∈ M′ is a ∗-isomorphism is satisfiedwhen M = B(H) and R = B(K). More generally, it is satisfied whenever thecentral projection of PH in R is IK (see Propositions 5.5.5 and 5.5.6 in [22]).Let (α,K,R) be an e-dilation of a semigroup Θ on M ⊆ B(H). (α,K,R) iscalled a minimal dilation if the central support of PH in R is IK and if

R = W ∗(⋃

s∈S

αs(M)).

Corollary 5.13. Let Θ be cp-semigroup on M⊆ B(H), and let (α,K,R) bea minimal dilation of Θ. Then Ξ(α) is an isometric dilation of Ξ(Θ).

5.3. cp-semigroups with no e-dilations. Obstructions of a new na-ture. By Parrot’s famous example [34], there exist 3 commuting contractionsthat do not have a commuting isometric dilation. In 1998 Bhat asked whether3 commuting CP maps necessarily have a commuting ∗-endomorphic dilation[10]. Note that it is not obvious that the non-existence of an isometric di-lation for three commuting contractions would imply the non-existence of a∗-endomorphic dilation for 3 commuting CP maps. However, it turns out thatthis is the case.

Theorem 5.14. There exists a cp-semigroup Θ = {Θn}n∈N3 acting on a B(H)for which there is no e-dilation (α,K,B(K)). In fact, Θ has no minimal e-dilation (α,K,R) on any von Neumann algebra R.Proof. Let T1, T2, T3 ∈ B(H) be three commuting contractions that have noisometric dilation and such that T n11 T

n22 T

n33 6= 0 for all n = (n1, n2, n3) ∈ N3

(one may take commuting contractions R1, R2, R3 with no isometric dilation asin Parrot’s example [34], and define Ti = Ri⊕1). For all n = (n1, n2, n3) ∈ N3,define

Θn(a) = Tn11 T

n22 T

n33 a(T

n33 )

∗(T n22 )∗(T n11 )

∗ , a ∈ B(H).Note that Θ = Σ(X,R), where X = {X(n)}n∈N3 is the subproduct systemgiven by X(n) = C for all n ∈ N3, and R is the (injective) representation thatsends 1 ∈ X(n) to T n11 T n22 T n33 (the product in X is simply multiplication ofscalars).Assume, for the sake of obtaining a contradiction, that Θ = {Θn}n∈N3 has ane-dilation (α,K,B(K)). Let (Y, V ) = Ξ(α) be the Arveson-Stinespring sub-product system of α together with the identity representation. By Theorem5.12, (Y, V,K) is a dilation of (X,R,H). It follows that V1, V2, V3 are a com-muting isometric dilation of T1, T2, T3 where V1 := V (1) with 1 ∈ X(1, 0, 0),V2 := V (1) with 1 ∈ X(0, 1, 0), and V3 := V (1) with 1 ∈ X(0, 0, 1). This is acontradiction.Finally, a standard argument shows that if (α,K,R) is a minimal dilation ofΘ, then R = B(K). �



Until this point, all the results that we have seen in the dilation theory of cp-semigroups have been anticipated by the classical theory of isometric dilations.We shall now encounter a phenomena that has no counterpart in the classicaltheory.By [49, Proposition 9.2], if T1, . . . , Tk is a commuting k-tuple of contractionssuch that

(5.5)

k∑

i=1

‖Ti‖2 ≤ 1,

then T1, . . . , Tk has a commuting regular unitary dilation (and, in particular,an isometric dilation). One is tempted to conjecture that if θ1, . . . , θk is acommuting k-tuple of CP maps such that

(5.6)

k∑

i=1

‖θi‖ ≤ 1,

then the tuple θ1, . . . , θk has an e-dilation. Indeed, if θi(a) = TiaT∗i , where

T1, . . . , Tk is a commuting k-tuple satisfying (5.5), then it is easy to constructan e-dilation of θ1, . . . , θk from the isometric dilation of T1, . . . , Tk. However,it turns out that (5.6) is far from being sufficient for an e-dilation to exist. Weneed some preparations before exhibiting an example.

Proposition 5.15. There exists a subproduct system that is not a subsystemof any product system.

Proof. We construct a counter example over N3. Let e1, e2, e3 be the standardbasis of N3. We let X(e1) = X(e2) = X(e3) = C2. Let X(ei + ej) = C2 ⊗ C2for all i, j = 1, 2, 3. Put X(n) = {0} for all n ∈ Nk such that |n| > 2. Tocomplete the construction of X we need to define the product maps UXm,n. Let

UXei,ej be the identity on C2 ⊗ C2 for all i, j except for i = 3, j = 2, and let

UXe3,e2 be the flip. Define the rest of the products to be zero maps (except the

maps UX0,n, UXm,0 which are identities). This product is evidently coisometric,

and it is also associative, because the product of any three nontrivial elementsvanishes.Let Y be a product system “dilating” X . Then for all k,m, n ∈ Nk we have

UYk+m,n(UYk,m ⊗ I) = UYk,m+n(I ⊗ UYm,n),

or

UYk+m,n = UYk,m+n(I ⊗ UYm,n)(UYk,m ⊗ I)∗,

and

UYk,m+n = UYk+m,n(U

Yk,m ⊗ I)(I ⊗ UYm,n)∗.



Iterating these identities, we have, on the one hand,

Ue3,e1+e2 = UYe3+e2,e1(U

Ye3,e2 ⊗ I)(I ⊗ UYe2,e1)∗

= UYe2,e3+e1(I ⊗ UYe3,e1)(UYe2,e3 ⊗ I)∗(UYe3,e2 ⊗ I)(I ⊗ UYe2,e1)∗

= UYe1+e2,e3(UYe2,e1 ⊗ I)(I ⊗ UYe1,e3)∗

(I ⊗ UYe3,e1)(UYe2,e3 ⊗ I)∗(UYe3,e2 ⊗ I)(I ⊗ UYe2,e1)∗,

and on the other hand

Ue3,e1+e2 = UYe3+e1,e2(U

Ye3,e1 ⊗ I)(I ⊗ UYe1,e2)∗

= UYe1,e3+e2(I ⊗ UYe3,e2)(UYe1,e3 ⊗ I)∗(UYe3,e1 ⊗ I)(I ⊗ UYe1,e2)∗

= UYe1+e2,e3(UYe1,e2 ⊗ I)(I ⊗ UYe2,e3)∗

(I ⊗ UYe3,e2)(UYe1,e3 ⊗ I)∗(UYe3,e1 ⊗ I)(I ⊗ UYe1,e2)∗.

Canceling UYe1+e2,e3 , we must have

(UYe1,e2 ⊗ I)(I ⊗ UYe2,e3)∗(I ⊗ UYe3,e2)(UYe1,e3 ⊗ I)∗(UYe3,e1 ⊗ I)(I ⊗ UYe1,e2)∗

= (UYe2,e1 ⊗ I)(I ⊗ UYe1,e3)∗(I ⊗ UYe3,e1)(UYe2,e3 ⊗ I)∗(UYe3,e2 ⊗ I)(I ⊗ UYe2,e1)∗.Now, UXei,ej were unitary to begin with, so the above identity implies

(UXe1,e2 ⊗ I)(I ⊗ UXe2,e3)∗(I ⊗ UXe3,e2)(UXe1,e3 ⊗ I)∗(UXe3,e1 ⊗ I)(I ⊗ UXe1,e2)∗

= (UXe2,e1 ⊗ I)(I ⊗ UXe1,e3)∗(I ⊗ UXe3,e1)(UXe2,e3 ⊗ I)∗(UXe3,e2 ⊗ I)(I ⊗ UXe2,e1)∗.Recalling the definition of the product inX (the product is usually the identity),this reduces to

I ⊗ UXe3,e2 = UXe3,e2 ⊗ I.This is absurd. Thus, X cannot be dilated to a product system. �

We can now strengthen Theorem 5.14:

Theorem 5.16. There exists a cp-semigroup Θ = {Θn}n∈N3 acting on a B(H),such that for all λ > 0, λΘ has no e-dilation (α,K,B(K)), and no minimale-dilation (α,K,R) on any von Neumann algebra R.Proof. Let X be as in Proposition 5.15. Let Θ be the cp-semigroup generatedby the X-shift, as in Section 2.3 of the paper. Of course, Θ, as a semigroupover N3, can be generated by three commuting CP maps θ1, θ2, θ3. X cannotbe embedded into a full product system, so by Theorem 5.12, Θ has no minimale-dilation, nor does it have an e-dilation acting on a B(K). Note that if Θ isscaled its product system is left unchanged (this follows from Theorem 2.6: ifyou take X and scale the representation SX you get a scaled version of Θ). Sono matter how small you take λ > 0, λθ1, λθ2, λθ3 cannot be dilated to threecommuting ∗-endomorphisms on B(K), nor to a minimal three-tuple on anyvon Neumann algebra. �



Note that the obstruction here seems to be of a completely different naturefrom the one in the example given in Theorem 5.14. The subproduct systemarising there is already a product system, and, indeed, the cp-semigroup arisingthere can be dilated once it is multiplied by a small enough scalar.

Part 2. Subproduct systems over N

6. Subproduct systems of Hilbert spaces over N

We now specialize to subproduct systems of Hilbert W∗-correspondences overthe semigroup N, so from now on any subproduct system is to be understoodas such (soon we will specialize even further to subproduct systems of Hilbertspaces).

6.1. Standard and maximal subproduct systems. If X is a subproductsystem over N, then X(0) = M (some von Neumann algebra), X(1) equalssome W∗-correspondence E, and X(n) can be regarded as a subspace of E⊗n.The following lemma allows us to consider X(m+ n) as a subspace of X(m)⊗X(n).

Lemma 6.1. Let X = {X(n)}n∈N be a subproduct system. X is isomorphicto a subproduct system Y = {Y (n)}n∈N with coisometries {UYm,n}m,n∈N thatsatisfies

Y (1) = X(1)

and

(6.1) Y (m+ n) ⊆ Y (m)⊗ Y (n).Moreover, if pm+n is the orthogonal projection of Y (1)

⊗(m+n) onto Y (m+ n),then

(6.2) UYm,n = pm+n

∣∣∣Y (m)⊗Y (n)

and the projections {pn}n∈N satisfy(6.3) pk+m+n = pk+m+n(IE⊗k ⊗ pm+n) = pk+m+n(pk+m ⊗ IE⊗n).Proof. Denote by UXm,n the subproduct system maps X(s)⊗X(t)→ X(s+ t).Denote E = X(1). We first note that for every n there is a well definedcoisometry Un : E

⊗n → X(n) given by composing in any way a sequence ofmaps UXk,m (for example, one can take U3 = U

X2,1(U

X1,1 ⊗ IE) and so on). We

define Y (n) = Ker(Un)⊥, and we let pn be the orthogonal projection from

E⊗n onto Y (n). pn = U∗nUn, so, in particular, pn is a bimodule map. For all

m,n ∈ N we have thatE⊗m ⊗Ker(Un) ⊆ Ker(Um+n).

Thus E⊗m⊗Ker(Un)⊥ ⊇ Ker(Um+n)⊥, so pm+n ≤ IE⊗m⊗pn. This means that(6.3) holds. In addition, defining UYm,n to be pm+n restricted to Y (m)⊗Y (n) ⊆E⊗(m+n) gives Y the associative multiplication of a subproduct system.



It remains to show that X is isomorphic to Y . For all n, X(n) is spanned byelements of the form Un(x1 ⊗ · · · ⊗ xn), with x1, . . . , xn ∈ E. We define a mapVn : X(n)→ Y (n) by

Vn(Un(x1 ⊗ · · · ⊗ xn)

)= pn(x1 ⊗ · · · ⊗ xn).

It is immediate that Vn preserves inner products (thus it is well defined) andthat it maps X(n) onto Y (n). Finally, for all m,n ∈ N and x ∈ E⊗m, y ∈ E⊗n,

Vm+n(UXm,n(Um(x)⊗ Un(y))

)= Vm+n

(Um+n(x⊗ y)

)

= pm+n(x ⊗ y)= pm+n(pmx⊗ pny)= pm+n

((VmUm(x)) ⊗ (VnUn(y))

)

= UYm+n((VmUm(x)) ⊗ (VnUn(y))

),

and (1.2) holds. �

Definition 6.2. A subproduct system Y satisfying (6.1), (6.2) and (6.3) abovewill be called a standard subproduct system.

Note that a standard subproduct system is a subproduct subsystem of the fullproduct system {E⊗n}n∈N.Corollary 6.3. Every cp-semigroup over N has an e-dilation.

Proof. The unital case follows from Corollary 5.10 together with the abovelemma. The nonunital case follows from a similar construction (where thedilation of a non-fully-coisometric representation is obtained by adapting [41,Theorem 4.2] instead of [43, Theorem 5.2]). �

Let k ∈ N, and let E = X(1), X(2), . . . , X(k) be subspaces of E,E⊗2, . . . , E⊗k,respectively, such that the orthogonal projections pn : E

⊗n → X(n) satisfypn ≤ IE⊗i ⊗ pj

and

pn ≤ pi ⊗ IE⊗jfor all i, j, n ∈ N+ satisfying i+j = n ≤ k. In this case one can define a maximalstandard subproduct system X with the prescribed fibers X(1), . . . , X(k) bydefining inductively for n > k

X(n) =

⋂

i+j=n

E⊗i ⊗X(j)

⋂

⋂

i+j=n

X(i)⊗ E⊗j .

It is easy to see that

X(n) =⋂

n1+...+nm=n

X(n1)⊗ · · · ⊗X(nm) =⋂

i+j=n

X(i)⊗X(j).



We then have obvious formulas for the projections {pn}n∈N as well, for example

pn =∧

i+j=n

pi ⊗ pj , (n > k).

6.2. Examples.

Example 6.4. In the case k = 1, the maximal standard subproduct systemwith prescribed fiber X(1) = E, with E a Hilbert space, is the full productsystem FE of Example 1.2. If dimE = d, we think of this subproduct system asthe product system representing a (row-contractive) d-tuple (T1, . . . , Td) of noncommuting operators, that is, d operators that are not assumed to satisfy anyrelations (the idea behind this last remark must be rather vague at this point,but it shall become clearer as we proceed). In the case k = 2, if X(2) is thesymmetric tensor product E with itself then the maximal standard subproductsystem with prescribed fibers X(1), X(2) is the symmetric subproduct systemSSPE of Example 1.3. We think of SSP as the subproduct system representinga commuting d-tuple.

Example 6.5. Let E be a two dimensional Hilbert space with basis {e1, e2}.Let X(2) be the space spanned by e1⊗ e1, e1⊗ e2, and e2⊗ e1. In other words,X(2) is what remains of E⊗2 after we declare that e2⊗e2 = 0. We think of themaximal standard subproduct system X with prescribed fibers X(1) = E,X(2)as the subproduct system representing pairs (T1, T2) of operators subject onlyto the condition T 22 = 0. E

⊗n has a basis consisting of all vectors of the formeα = eα1 ⊗ · · · ⊗ eαn where α = α1 · · ·αn is a word of length n in “1” and “2”.X(n) then has a basis consisting of all vectors eα where α is a word of lengthn not containing “22” as a subword. Let us compute dimX(n), that is, thenumber of such words.Let An denote the number of words not containing “22” that have leftmostletter “1”, and let Bn denote the number of words not containing “22” thathave leftmost letter “2”. Then we have the recursive relationAn = An−1+Bn−1and Bn = An−1. The solution of this recursion gives

dimX(n) = An +Bn ≈(

1 +√

5

2

)n.

As one might expect, the dimension of X(n) grows exponentially fast.

Example 6.6. Suppose that we want a “subproduct system that will representa pair of operators (T1, T2) such that TiT2 = 0 for i = 1, 2”. Although we havenot yet made clear what we mean by this, let us proceed heuristically alongthe lines of the preceding examples. We let E be as above, but now we declaree1⊗e2 = e2⊗e2 = 0. In other words, we define X(2) = {e1⊗e2, e2⊗e2}⊥. Onechecks that the maximal standard subproduct system X with prescribed fibersX(1) = E,X(2) is given by X(n) = span{e1⊗ e1⊗ · · · ⊗ e1, e2⊗ e1⊗ · · · ⊗ e1}.This is an example of a subproduct system with two dimensional fibers.



At this point two natural questions might come to mind. First, is every stan-dard subproduct system X the maximal subproduct system with prescribed fibersX(1), . . . , X(k) for some k ∈ N? Second, does dimX(n) grow exponentiallyfast (or remain a constant) for every subproduct system X? The next exampleanswers both questions negatively.

Example 6.7. Let E be as in the preceding examples, and let X(n) be asubspace of E⊗n having basis the set

{eα : |α| = n, α does not contain the words 22, 212, 2112, 21112, . . .}.Then X = {X(n)}n∈N is a standard subproduct system, but it is smaller thanthe maximal subproduct system defined by any initial k fibers. Also, X(n) isthe span of eα with α = 11 · · ·11, 21 · · ·11, 121 · · ·11, . . . , 11 · · · 12, thus

dimX(n) = n+ 1,

so this is an example of a subproduct system with fibers that have a linearlygrowing dimension.

Of course, one did not have to go far to find an example of a subproductsystem with linearly growing dimension: indeed, the dimension of the fibers ofthe symmetric subproduct system SSPCd is known to be

dimSSPCd(n) =

(n+ d− 1

n

).

Taking d = 2 we get the same dimension as in Example 6.7. However, SSP :=SSPC2 and the subproduct system X of Example 6.7 are not isomorphic: forany nonzero x ∈ SSP (1), the “square” USSP1,1 (x ⊗ x) ∈ SSP (2) is never zero,while UX1,1(e2 ⊗ e2) = 0.Here is an interesting question that we do not know the answer to: given asolution f : N→ N to the functional inequality

f(m+ n) ≤ f(m)f(n) , m, n ∈ N,does there exists a subproduct system X such that dimX(n) = f(n) for alln ∈ N?Remark 6.8. One can cook up simple examples of subproduct systems thatare not standard. We will not write these examples down, as we already knowthat such a subproduct system is isomorphic to a standard one.

6.3. Representations of subproduct systems. Fix a W∗-correspondenceE. Every completely contractive linear map T1 : E → B(H) gives rise to a c.c.representation T n of the full product system FE = {E⊗n}n∈N by defining forall x ∈ E⊗n and h ∈ H(6.4) T n(x)h = T̃1

(IE ⊗ T̃1

)· · ·(IE⊗(n−1) ⊗ T̃1

)(x⊗ h),

where T̃1 : E ⊗ H → H is given by T̃1(e ⊗ h) = T1(e)h. We will denotethe operator acting on x ⊗ h in the right hand side of (6

subproduct systems - uni-bielefeld.de · 2010. 1. 15. · orr moshe shalit and baruch solel 1...

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